Shakib Al Hasan, Mushfiqur Rahim and Najmul Hossain all hit fifties but England's bowlers restricted Bangladesh to 246 all out in the third one-day international in Chittagong on Monday.
Needing a win to avoid a whitewash in the three-match series, the hosts elected to bat first but were bowled out in 48.5 overs.
Shakib made 75 and Mushfiqur and Najmul added 70 and 53 respectively.
Jofra Archer picked up 3-35, while Adil Rashid and Sam Curran were the other two stand-out bowlers with 2-21 and 2-51.
Curran struck early, handing opener Liton Das a duck in the very first over before he dismissed Tamim Iqbal for 11.
Najmul and Mushfiqur shared a 98-run partnership for the third wicket to steady Bangladesh's innings before the former was run out by Jos Buttler.
Rashid bowled Mushfiqur and Archer ended Shakib's 71-ball innings in the penultimate over, wrapping up Bangladesh's innings the next ball.
England handed an ODI debut to 18-year-old leg-spinner Rehan Ahmed, who finished with a modest 1-62.
England lead the series 2-0 after winning the first two matches respectively by three wickets and 132 runs.
It is England's last scheduled ODI until September 8.
The ODI series is crucial to preparations for the World Cup in India later this year.
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